Retrospective Theses and Dissertations Iowa State University Capstones, Theses andDissertations

1960

Magnetic properties of rare earth metalsSamuel Hsi-peh LiuIowa State University

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LIU, Samuel Hsi-peh. MAGNETIC PROPERTIES OF RARE EARTH METALS.

Iowa State University of Science and Technology, Ph.D., 1960 Physics, solid state

University Microfilms, Inc., Ann Arbor, Michigan

Approved:

MAGNETIC PROPERTIES OF RARE

EARTH METALS

by

Samuel Hsi-peh Liu

A Dissertation Submitted to the

Graduate Faculty in Partial Fulfillment of

The Requirements for the Degree of

DOCTOR OF PHILOSOPHY

Major Subject: Physics

In Charge df lajor Work

Head of Major Department

Dean 6. Gi'âduat Çollege

Iowa State University Of Science and Technology

Ames, Iowa

I960

Signature was redacted for privacy.

Signature was redacted for privacy.

Signature was redacted for privacy.

ii

TABLE OF CONTENTS

Page

INTRODUCTION 1

REVIEW OF EXISTING THEORIES L

THE S-F INTERACTION 20

INTERPRETATION OF MAGNETIC PROPERTIES OF DYSPROSIUM 37

BIBLIOGRAPHY 63

ACKNOWLEDGEMENT 66

1

INTRODUCTION

The rare earths form an especially interesting group of elements

in the periodic table. For the most part they have three electrons in

the valence shell or conduction band, and as the atomic number increases

the inner l|f shell is gradually filled up. The existence of an unfilled

k£ shell leads to many interesting features in the electric and magnetic

properties of these metals. In particular there is a smooth dependence

of these properties on the number of electrons in this shell.

Most of these metals are paramagnetic at room temperature.

Gadolinium and terbium become ferromagnetic below 289° K and 230° K

respectively. Dysprosium, holmium, erbium and thulium also exhibit a

transition to an ordered magnetic state when the temperature is lowered.

This state is known to be antiferromagneti c and to have a very compli­

cated structure. At still lower temperatures they undergo another

transition and become ferromagnetic. Cerium, neoctymium, samarium become

antiferr omagneti c at about 10° K (Lock, 19$7). No ordered state has

been observed in other elements of the group. A collection of pertinent

data is found in the review article of Spedding et. al. (1957).

The magnetic properties of these metals are due primarily to the

electrons in the incomplete shell. In fact the magnetic moment per at cm

is determined by the number of electrons in the l|f shell according to

the Russell-Saunders coupling scheme. However, one needs to consider

other effects in order to understand the ferromagnetic or antiferro-

magnetic coupling in Gd etc. In these metals the Ijf shell of each ion

is completely surrounded by the $s and 5p shells. As a result the

2

direct exchange effect between neighboring ions must be very small

because their magnetic shells have very little chance to overlap each

other. Therefore the strong coupling in Gd etc. must be due to other

mechanisms. One such mechanism which is believed to be the most important

is the indirect exchange coupling via conduction electrons. This inter­

action can roughly be described as follows. A conduction electron first

interacts with the l^f shell electrons of an ion through the electrostatic

Coulomb exchange effect. This tends to line up the spin of the conduction

electron parallel or antiparallel to the spin of the ion. As the

electron travels in the crystal it also interacts with other ions and

thus gives rise to a force which tends to line up all the ions in the

crystal. Therefore a long range type of coupling results.

The interaction between the conduction electrons and the l*f shell

electrons (s-f interaction) is believed to be responsible for many other

properties of rare earths as well. Considerable success has been

achieved in interpreting the anomalous resistivity of some rare earths and

the reduction of superconductive transition temperature of dilute

solutions of rare earths in lanthanum by this interaction.

Most of the rare earth metals have the hexagonal closed packed

structure. Weutron diffraction experiments on holmium and erbium have

revealed their very complicated magnetic structures when they are anti-

ferromagnetic*. In holmium it is found that the best fit to the

experimental data is given by a spiral structure. The ion spins in

*W. C. Koehler, Oakridge National Laboratory, Oakridge, Tenn. Infor­mation concerning the magnetic structure of erbium and holmium metals. Private communication. I960,

3

each basal plane are parallel and are perpendicular to the c axis. The

resultant magnetic moments of adjacent planes make a certain angle such

that a staircase structure is formed along the c axis of the crystal.

The angle between adjacent planes is about 38° at lt5° K and increases

linearly with temperature up to 1|8.5° at 119° K, the Neel temperature

being 132° K. Dysprosium, erbium and thulium may have similar structures

when antiferromagnetic. The origin of the spiral structure as well as

that of the ferromagnetic to antiferr omagneti c transition has been the

theme of many discussions.

The present work consists of two parts. The first part contains a

detailed derivation of the first order s-f interaction Hamiltonian using

basic principles. Some of the consequences of this interaction are also

discussed. The second part is a phenomenological interpretation of the

magnetic properties of dysprosium single crystals. Detailed agreement

for the magnetization curves in the ferromagnetic and antiferromagnetic

regions is obtained.

h

REVIEW OF EXISTING THEORIES

The model for rare earth metals one usually employs consists of a

lattice of trivalent ions in a sea of conduction electrons. This model

describes all members of the group except the following: cerium has

four conduction electrons at low temperatures; europium is divalent;

and ytterbium is divalent and has a filled Uf shell. The element

promethium is radioactive, so very little is known about its physical

properties. The present discussion of rare earth metals will exclude

these exceptional cases.

The angular momentum and the magnetic moment of a trivalent rare

earth ion are entirely determined by the structure of the Uf shell. It

has been established that the electrons in this shell couple their

orbital and spin angular momenta together according to the Russell-

Saunders LS-coupling scheme. (Van Vleck 1932). The orbital angular

momenta of the electrons couple into L and the spin angular momenta

into S by the electrostatic interaction. Then L and S couple into the

total angular momentum J by the spin-orbit coupling. For a free ion Jz

and are constants of motion of the dynamic system. In a metal the

crystalline field splits levels of different Jz, however for most

members of the group this effect is quite small compared with the

multiplet splitting, the splitting of levels of the same L, S but

different J. In a first order theory the ions are usually treated as

free. The crystalline field splitting may give rise to magnetic

anisotropy (Niira, i960).

5a

In the presence of a magnetic field H, the Hamiltonian of the

interaction between the l|f shell of an ion and the field is given by,

in the units of C = = 1,

H ' = - Z- (e/2m)H.(r^ x %) _ J (e/m)ïF*sJ, (l)

where e is the charge of an electron, r^, pj_ and Sj_ are the position,

momentum, and spin vectors of the i-th electron, and the summation is

taken over all electrons in the shell* By definition of L and S

operators, Hf can be written as

H' = (lel/2m)H»(L + 2S) .

Taking H in the 2-direction, one obtains

H» = (|e|H/2m)(Lz + 2SZ)

= ([e|H/2m)(Jz +SZ) . (2)

The matrix element of the interaction is, to the first order,

<jji ( h' 1 jjz jjZ 1 jz+Szijjz •

5b

By the projection theorem of angular momentum, this is equal to

-| JJZ>= < «. | (i + jt | >

= Lsii fx + J(J+i)-L(L+i)-ts(s+i)] 2m *- 2J( J+l) >

XJ2 £" JZJ« . (3)

The quantity in the square bracket is called the Lande g-factor and is

denoted by g. The quantity is the Bohr magneton Therefore the

degeneracy in J2 is completely split and the energy levels are equally

spaced,

Elz * H • (i«)

The magnetic moment operator is defined by

H ' = - p ' K . (5)

> Therefore, with H in z-directicm

6

F"* = -js(1»+2s,)

" 5" (Jz + Sz> • (6)

The expectation value of pz in the state IJJ2 > is therefore

^ z^= " ^sKJJzlJz + sJJJs>

~ - H B SJz • (7)

For a system of N free ions per unit volume in thermal equilibrium,

the energy levels of different Jz are occupied according to the Boltzmann

distribution. The total magnetic moment of the system is therefore

is the Brillouin function. This gives the magnetic moment of a collection

of free ions as a function of temperature and applied field.

In many rare earth metals the spins of the ions are coupled

together ty ferromagnetic or antiferrcmagnetic coupling. The simplest

model for a ferromagnetic crystal is one -where the coupling between spins

is represented by the Hamiltonian

M = )exp(-p(Jz)

1 Jz exp(-Of J2)

( 8 )

and B (a ) = J * V2 V

(9)

Hex - -2 Aij jj/Jj • (10)

Aij represents the exchange energy between the i-th and the j-th ions.

This Hamiltonian can be written alternatively as

Hex = -2i [Ij Jj ] * Ji

Therefore the coupling can be represented by an effective field acting

on the ions. For the i-th ion, the effective molecular field is

= iki

Is 413 Ti " tài

Alj < 7 > (n)

where ^ denotes the average spin over a certain number of nearest

neighbors of the i-th ion. In the molecular field approximation, one

approximates

"*/ H B & •

So the molecular field acting on the i-th ion is

X. = irrV" ™ = A ™ • (12)

r B G N

with

• (13)

A is called the molecular field constant. Hence for a ferromagnetic

crystal, the total effective field is the sum of the external field and

8

the molecular field,

H = H 0 + H m = H 0 + M . ( I l l )

Substituting Equation (lit) into Equation (8), one obtains an implicit

dependence of the magnetic moment M as a function of the external field

and temperature.

If there is no external field applied, M as a function of temperature

is given by

M = N gJB (15) D kT *

Equation (15) has a non-zero solution for M as long as the

temperature is less than a critical value Tc, known as the Curie

temperature, given try Tc = 2g2NJ(J+l) ^ (16) 3k

From Equation (13), one obtains

*. = I Aid , (17)

which gives the relationship between the ferromagnetic-paramagnetic

order-disorder transition temperature and the strength of coupling

between ions.

At a temperature T > Tc ,, the system has no net magnetization unless

an external field is applied. The relationship between M and K0 is given

by Equation (8) with H = H0 + 7, M. At high enough temperatures the

Brillouin function can be approximated by

9

B (ex) = d/3 (J + 1) .

Thus one obtains

M 1*1 - N ^B^JtJ+l) 1 = N fVe2J(J+l) „ C 3kT J 1PT o •

Hence the crystal is paramagnetic with the paramagnetic susceptibility

*= H = rrs (is)

•where

= • "Ww) (19)

is the Curie constant.

For large enough J, the Brillordn function can be approximated by

the classical Langevin function

X (x ) = cot x - | . (20)

It can be shorn that

B X (41")

•where

H = H B g\/j («J +1). (21)

The quantity g J J( J+l) is usually called the equivalent number of Bohr

magnetons of the ion.

10

An analogous theory of antif err omagneti sm was derived by Néel

(1932, 1936). In thts theory one considers two identical sublattices

A and B with self and mutual interactions. To illustrate the content

of the theory a special model will be considered. Assuming that the

sublattices have ferromagnetic interaction within the sublattices and

antif err csnagneti c interaction between them, one may write the effective

fields acting on the sublattices as

Ha = Ho + ^ ,

HB = Ho - a 'ÏA + a Mg . (22)

The applied field is assumed to be in line with M^, Mg. Therefore M^,

Mg are solved from

Ma = 1/2 HB gNJB BgHAj t

Mb" = 1/2 Hb . (23)

With no external field applied, one can find non-zero solutions for %,

Mg provided that the temperature is lower than T^, the Neel temperature,

given by

% = * ^ ( 2 1 4 )

where /V is defined by Equation (21). At high enough temperatures the

crystal becomes paramagnetic with the susceptibility

11

X " ÏTÏ7 (25) N

•where

TjJ = (a-K*') NH2 (26) ti 6k

and C is the Curie constant in Equation (19). The susceptibility in

Equation (25) is usually called the parallel susceptibility. When the

field is perpendicular to the magnetic moments one needs to consider other

effects in order to study the behavior of the crystal.

Along the same line as the molecular field approximation, Neel

(1956 a,b) gave an explanation to the ferromagnetic to antiferro-

magnetic transition of dysprosium. He assumed the material to consist of

two sublattices with strong ferromagnetic exchange forces within each

sublattice and weak interactions between them. The intralattice exchange

forces are so strong that they are the only significant terms in the

effective field expressions (22). Therefore to the first approximation

M^, Mg are regarded as independent of the applied field as well as their

relative orientation. The interlattice interactions are important in

determining the ordering of the sublattices. When a field is applied,

one writes the total energy E as a sum of the interaction with the

applied field

Eh = -H * (% + Mg ) ,

an exchange energy between the sublattices

12

E„- = + L/ltnM2 cos (0A - ©B) >

and a magnetocrystalline energy

Ec = - 1/2 K0 (cos2 64 + cos2 eg) - Klcos 0A cos % • (27)

0 , 6g are the orientations of the sublattice magnetic moments with

respect to the axis of magnetization and 1/2 M = | % | = | % | . The

magnetocrystalline energy is the energy associated with the axis of

magnetization. The equilibrium configuration, and thus the magnetization

curve, is found by minimizing the total energy with respect to %

6B.

The properties of the system depend on the relative sizes of the

parameters as discussed in detail by Neel. With no external field

applied, the equilibrium state of the system is antiferromagnetic

(% = 0, ©b ~7r)» if K0 - Ki>0, Ki < l/ljnll2 and is ferromagnetic

(0A = 6g = 0) if K0 - K]_ > 0, K]_ > l/k nM2. Therefore, a transition

from ferromagnetic state to antif err cmagnetic state is possible if K%

and nH2 have different temperature dependences. It is also of interest

to study the magnetization curve when the system is antiferromagnetic.

18hen the field is in the direction of the axis of magnetization, one

finds that if K0 £ l/lpiM2 the system remains antiferromagnetic as long

as H is less than a critical value Hs given by

Hs = l/2nM [ 1 - to- ] . nl(2 '

"When H becomes greater than Eg the system goes into the ferromagnetic

13

state with a magnetic moment M. There is a discontinuity in the

magnetization curve at H = Hs. This resembles roughly the observed

magnetization curves for dysprosium in the temperature range from 85° K

to 179° K.

The origin of the ferromagnetic coupling in rare earth metals is

also a problem of great interest. It is apparent that the strong

ferr (magnetic coupling is not a result of the direct exchange inter­

action of Heisenberg because there is a lack of overlap between

neighboring JUf shells. Kasuya (1956a) was the first to point out that

this paradox could be resolved by the Zener's model of ferromagnetism

(Zener, 1951; Zener and Heikes, 1953)» namely the indirect exchange

coupling via conduction electrons. In his paper, Kasuya studied in

some detail the exchange interaction between a conduction electron and

the magnetic shell electrons of an ion, and the indirect exchange

coupling between different ions resulting from this s-f interaction.

For gadolinium where there is no spin-orbit coupling, he showed that the

scattering of a conduction electron by an ion through the s-f inter­

action can be represented by an operator

where a^creates an electron with momentum k and spin up, ak»| destroys

—»• -*> —>

an electron with momentum kT and spin down etc; k',k are the initial and

final momenta of the conduction electron; R is the position and S is the

spin of the ion, SÎ = S%+ iSy, A (k,k* ) is the strength of interaction.

•1/2 A (k,k« ) [(a£fa'k,f - akjaktj )S2+ akfaktj S

i

iu

In terms of spin operators, it can be written simply as

H' = -A(k,k' )s.S exp [i(k -k1 )Ȕf], (28)

where s is the spin operator of the conduction electron. In

gadolinium S is equal to the total angular momentum J, but for other

rare earth metals the spin-orbit coupling complicates the problem.

De Germes (1958) however suggested that the Hamiltonian in Equation (28)

should be applicable to all rare earths if S is taken as the spin

angular momentum of the ion. Since J is the exact quantum number of

the ion, one may consider only the interaction within the manifold of

ground state J. Then S can be replaced by its projection on J and

where g is the Lande factor. A fundamental derivation of the

Hamiltonian (29) will be given in the text of the present work. It will

be shown that under certain approximations the Coulomb exchange inter­

action between a conduction electron and an ion can indeed be represented

by this Hamiltonian. Furthermore the strength of interaction A (k,k* )

depends on the conduction band structure of the metal and is independent

of the directions of k,k'.

Following Ruderman and Kittel:s (195U) model of indirect exchange,

one can show that the interaction in Equation (29) leads to an inter­

action Hamiltonian between ions of the form in Equation (10). It is

assumed that the interaction between the spins Jj. and Jj located at Ri

and Rj arises from a double scattering of a conduction electron. By the

H» = - A (k,kl) (g-l)s-J exp [i(k-k')*R ] , (29)

15

second order perturbation theory the energy of interaction is

Hjj - -(s»J^)(s»Jj)

x

1

(g-l)2|A(k,k')!2 exp[i(k -4c')*(Ri -Rj)J

mn—- e(5)

+ complex conjugate (30)

The first term represents the coupling which arises when the electron

is first scattered by the i-th ion and then by the j-th ion, the second

term, Tihich is the complex conjugate of the first, arises from a

scattering process of the reversed order, denotes the wave number

corresponding to the Fermi energy. The second integral extends between

the limits and infinity because all levels below are filled, so

by the Paull principle k1 must take on seme value greater than 1%. The

largest contribution to the integrals comes from values of k,k' near

the Fermi energy, so one may approximate | A (k,k' ) | 2 by

| A (]%,k^)|2 and take it outside the integral signs. Summing over the

spins of the electron, one reduces Equation (30) to

Here Rjj = Rj_ - Rj. If one assumes an isotropic energy band structure for

Hj_j - - M | 2 (g-lj

+ complex conjugate (31)

16

the conduction electron such that

E(k) = "fi2k2/2m* ,

then the integral in Equation (31) can be readily evaluated. The final

result as given by Ruderman and Kittel is

Therefore, one obtains Equation (10) if one sums Hj_j in Equation (32)

over all ions. The strength of the indirect exchange coupling

is an oscillatory function of the distance between ions and is of much

longer range than the direct exchange coupling.

Most rare earths have very similar physical properties. Accordingly

de Gennes (1958) assumed that the strength of s-f interaction A (k%,k^)

is the same for all members of the group. If the metals have similar

conduction band structures, then the free electrons should have the

same effective mass m*. Hence one may conclude that

Hij " -2Aij » (32)

where

A. , = _ | A (km,W I 2(s -1)2

(U 7r )3ii2

(~7)^(2kBPijcos(2iyiij) - sin(2k^ij )J X J

Ajj PC (g -1)2 (3D

Substituting ,his into the expression for the Curie temperature

17

Equation (16), one obtains

le"1 (g -1)2J(J + 1) . (35)

This is the de Gennes formula for paramagnetic Curie temperatures of

rare earth metals. The same result was also obtained by Brout and Suhl

(1959)* For the elements gadolinium to lutetium there is the

relationship J = L + S, so

Tcoo S2(J +i)/j (36)

which is the Neel formula. This result is in agreement with the observed

paramagnetic Curie temperatures of these elements. For the elements

lanthanum to samarium J = L - S applies, so

Tc°^ S2J/(J + 1) . (37)

This however does not agree with the experiments.

The long range nature of the indirect exchange coupling may give

rise to the peculiar magnetic structure of some of the rare earths. For

the study of this effect Villain (1959) proposed a very interesting way

of investigating the relative stability of various magnetic structures.

Under certain conditions the most stable configuration may be a spiral

structure.

From Equation (11) the molecular field at the i-th ion is

Hmi = Z jAj_j Jj/ p Bg *

The average value of Jj_ is given by the equation

18

h = jAijJ^kT) (38)

where u^ is the unit vector in the direction of the molecular field.

Near the critical (or Neel) temperature one has I j «J. The

Brillouin function can be expanded to the first order, so

Ji = J(J +1)1 /i//3kT . (39)

If one defines the following Fourier transforms

T(k) = Z iJiexp(ik.Ri) ,

f(k) = Z j^ijexp [ik.(% - Rj)] ,

Equation (39) can then be written as

~T(k) = J 1 } 5(k)T(k) .

Thus one obtains the Neel temperature % as

% = j(jjy) ?<ïo) (M»

where ICQ is the k -which gives a maximum value to Ç (k)% In this way

the critical temperature is obtained by a study of the reciprocal lattice

structure instead of the various possible sublattice structures. In the

reciprocal lattice space the possible points k0 form a set which is

invariant under translation, inversion at the origin and the symmetry

operations of the reciprocal lattice.

Assuming for simplicity that there exist only two vectors k0 which

19

are symmetric with respect to the center of the lattice, one can show

direct substitution that Equation (39) has solutions of the form

where ci — x, y, z. This is seen to be a spiral structure along the k0

direction. In a hexagonal lattice it is possible to find such a pair

of k0 directions only along the c axis. Therefore, if such spiral

structure exists in a crystal with hexagonal lattice, it must have its

spiral axis along the c direction. This has been verified experimentally

in holmium and erbium*. Villain also proved that a spiral structure of

this kind is stable below the Neel temperature by showing that any small

deviation from this arrangement tends to raise the free energy of the

system. The magnetostriction effect is neglected in this analysis.

According to this theory the spiral angle is determined completely

by the strength of coupling Ajj and the structure of the reciprocal

lattice. Consequently the spiral angle should be insensitive to

temperature. This conclusion, however, is not in agreement with the

observation in holmium. There it was found that the spiral angle is

I4.8.50 at 110° K and decreases linearly with temperature until at U5° K

it becomes 38°.

C. Koehler, Oakridge National Laboratory, Oakridge, Tenn. Information concerning the magnetic structure of erbium and holmium metals. Private communication. 1960.

(ia)

20

THE S-F INTERACTION

In this section Kasqya's work (1956 b) is extended to other rare

earth metals where the spin orbit coupling must be taken into account.

It will be proved that the Coulomb exchange interaction Hamiltonian

can be put in the form of Equation (28) under certain approximations.

This gives a fundamental proof of de Gennes* proposal (1958).

Basic Model and Wave Functions

In this analysis one is interested in the electrostatic Coulomb

forces between a conduction electron and the core electrons of an ion.

Since the filled shells of the ion do not contribute to any angular

momentum, it is sufficient to consider only the i|f shell electrons of

the ion. The interaction Hamiltonian may be written as

N «X =2 «1 (b2)

i l?i -FN+1|

where

^N+l = coordinates of the conduction electron,

r^ = coordinates of the i-th magnetic shell electron.

The summation is taken over all electrons in the shell. The

perturbation method will be used to calculate the interaction matrix

elements. The appropriate wave function for the conduction electron is

of the form

Y'(r^s) = ug(r)exp(ik*r) X (1*3)

21

which is the Pauli wave function for a Bloch wave normalized in a large

volume. 1( is the Pauli spinor. The wave function for the magnetic

shell has a rather complicated structure. In absence of spin-orbit

coupling and electrostatic interactions each electron in the h£ shell

would have a wave function of the form

B(r)Y^( »,t) x • (to)

The residual interactions couple the electrons together according to

the Russell-Saunders scheme, i.e., the orbital angular momenta of the

electrons couple into L, spin angular momenta couple into 5, and then

L and S couple into J. So

—9 —»

J = L + S

L = 2

S =2?; .

The wave function of the shell will be of the form

YjK(l,2,-',i,'"N) = Zc(LSj;m,M - *0^(1,2, —H)

XVs,M-m(1'2'**#K)- ^

The wave function Tjn anà 3jre ®iëen functions of the operators

* p L , Lz and S , Sz respectively, and are constructed according to the

Pauli principle and Hund's rule of maximum multiplicity. Two cases

will be discussed separately.

When the magnetic shell is less than half or half filled

(N ^ 2 JL +1), V's,M-m *s completely symmetrical and is completely

22

antisymmetrical with respect to exchange of particles. The electrons

must be in different eigenstates of the operator Jlz . According to

Hund's rule

S = N/2,

L = Ni - N(N - l)/2,

J = L - S = N^ - N2/2 for 05 N S 2i,

= L + S = JL + 1/2 for N = 2 Â + 1.

Therefore the spin function V's,M-m has the form of a symmetrized product

^S,M _ = jjjfoc(l) (2)«- « (p)

x ft (p + !)•••• (N)J (1|6)

where j>J is the symmetrization operator defined by

J = 2 ?,

P is a permutation operator of the N particles and the sum is taken over

all possible permutations. oc, {3 are the spin up and spin down eigen-

functions; A is the normalization factor which is equal to J N I p 1 (N - p ) 1 j and p = S + M - m. The space function contains a linear combination of

products of single particle wave function (r), and is completely

antisymmetrical with respect to all N particles. It is convenient for

later calculations to group the terms together according to different

quantum states of the i-th particle, i.e.

23

Y im(l,2,""N) = Z 3^(1,2,...,! - l,i + 1,—N)

• (U7)

The normalization property of tjA Lm assures that

H- ' ^ J " r

dr^* • *dri_idr^+1* • -drN =1 . (U8)

Putting (It6) and (It?) into (li5), one obtains the wave function for the

magnetic shell.

TShen the magnetic shell is more than half filled (N > 2 X + l),

the space and spin wave functions have more complicated symmetries. It

is most convenient to express the symmetries by Young diagrams shown

in Figure 1 (Landau and Lifshitz, 1956, pp. 210-21$). One symmetrizes

with respect to all particles in the same row and then antisymmetrizes

•with respect to all particles in the same column of the diagrams. Let

there be 2q electrons that are paired off and n electrons that are

unpairedj then Hund's rule gives

the Young diagrams such that the first (2 Jl +1) electrons are in the

long column of the space diagram and in the long row of the spin

S = n/2,

L = nJl -n(n-l)/2,

J — L + S = n - n(n—2 )/2 .

If one labels the electrons by 1, 2, , N and arranges them in

2h

SPACE DIAGRAM

SPIN DIAGRAM

Figure 1. The Young diagrams for space and spin symmetries of a more than half filled lif shell#

25

diagram, one can carry out the symmetrization and antisymmetrization

and obtain the wave functions

V" LMjtC1»2»'*^) and V/s,MHnt(1>2»,,*N) ,

where t denotes the complementary tableaux obtained by this arrangement

of the particles in the frames. Similar terms can be formed by arrang­

ing the particles in different ways in the same frames. The completely

antisymmetric eigenfunction of the operators L2, Lz, $2 and Sz can then

be expressed by a sum of the farm

Z V-i^tOL.S,—N) «) (to)

•«here the summation is taken over all possible tableaux with the same

frame. One should note that since the terms in the above sum are not

linearly independent, the coefficients are not uniquely defined

even though the sum is. Therefore the required eigenfunction of J2

and Jz is

"/'JM = I C(LSJ;m,M-m)ilntVl1„,ty's,M-m,t • (#) m,t

Matrix Elements of Interaction

Now one can calculate the exchange part of the matrix elements of

the interaction Hamilton!an (U2), using a wave function constructed

26

from (U3) and (U5)« The wave function of the system of one conduction

electron and one magnetic shell is, with no regard to symmetry,

^ Y(%+1) (51)

where V" (N+1) denotes (rN+iJsN+l) of the conduction electron. In

general this wave function should be antisymmetrized with respect to all

(N+l) electrons. However, one may avoid this tedious procedure by

making the following observations. The Hamilton!an (ij2) contains a sum

of two particle operators and is completely symmetrical with respect to

the magnetic shell electrons. It can be proved that one obtains the

correct energy if one calculates the contribution of each term of

Equation (ij.2) separately using a wave function which is antisymmetrized

only with respect to the two particles involved. As an example, for

the term —Sf , the proper wave function may be taken as

l^i"*N+l'

i = hfy JM(l,2,—i,-'N)Y'(N+l) v/ 'i *•

- Y-JM(1>2,"-N+1,"-N) V" (i)J . (52)

Assuming the following initial and final states (unsymmetrized)

5i = V-JMU»2,...N)V(N+1) ,

5f = VjM'(l,2,'"N) f'(N+D ,

where

V'(N+l) = ug(i^+1)exp(ik*r^+;L)XN+1 ,

27

^ 1 (N+l) - uk,(rN+l)exP(ik'*rN+l) X N+l ,

2 one finds the contribution of the term —— to the matrix

I ri_rN+ll element to be

>2 % = V-'*(N+iw®

' |ri -^r+xl

* "Vtitt1'2, • • •!» • • -N) "V- (B+l•• a +1

-Jf^,(1,2,"-i,—N)V'*(N+!)[=-? 2

ri "*rN+l'

x ' '-N+:L' * ' eN) Y' (i)di dr • • 'df +i (53)

after seme simple manipulations. The second term is the exchange part

one is interested in. For the entire shell the exchange part of the

matrix element is

Hex = -Zi JVjii't1'2»*"1»"'1*) V''*(H+1)|Yi ! +1|

x e,,N) y,(i)dri*"dfÇ+1 . (Sh)

This "will be calculated separately for N 2 A +1 and N > 2 Ji +1.

In the case of N ^ 2 A +1, each term of (5W contributes the same

amount because of the symmetry of the space functions. So

= - M/f I

X ^(1,2,...N+l,... N)^(i)d^- d^+1 .

28

Putting in the wave function (U5), one obtains

Mex = -N% C(LSJ;m,M-m)C(lSJ;m%M'-m')

* yV' Im'C1»2»*** iv iV'N)

x 'HN+1)JT=k i {1'2i * * -N+1' * *,N )

x S,M-m(1J2***'N+1*'"N) V' (i)^i**dr^+i (55)

In each term of (55) the spin functions are multiplied together to form

a sum of products of Pauli spinors for all the particles. A typical

term in the sum is of the form

X ]*(!) y~ g*(2)" X ~X H+I(N+1)

* XjU) X 2(2)-^ N + 1(i)» AT hCN) AT i(H+l) ,

•which is zero unless

X j = X j jàîior N+l

and X i = N+l , ^ N+l = ^ i • (56)

If these are satisfied the product equals unity. The following four

cases will be examined.

1. X N+1 = X N+l = ^ • In this case X j ~ X j for

1 é j < N. Hence M-m = M'-m'. The total number of non-zero terms

29

in the product of spin function can be found to be p»p 1 (N-l) i (N-p) ! .

The product of the spin functions is this quantity divided by the

product of the normalization factors. Thus it is found to be p/ïî and

(^ex)l = - C(LSJ$m,M--ni)C(LSJjm',M,-m')(S-fiI-m)

x Im,^,2j " "i, * "N)u^, (r^^)exp(-ik' "%^)

e2 X IF. -r T - I ^ 2' ' '*N+1> 'e *N )

i N+l1

x exp(ik.f:)d5%-. dr^+-|_ % (57.1)

after putting p = S + M - m.

2. X Q-KL = J( jj+i = P . In this case the condition M-m = M'-m1

still holds. The product of the spin functions comes out to be

(N-p)/k = (S-M-hn), so

(MeX)2 = - Z C(L5J;m,M-4n)C(LSJ;m',M'-m' )(S-M-#m) mm'

x(same integral) % %_m,M'-m' . (57.2)

3» X N+l = P 3 ^ N+l = • In this case X ± ~ >

X i = 0{ , so M-m = M'-m'+l. The product is

30

V- SjM'-m' X NÏ1 Vs.ÏHn I N+l = i/ptS-p+l)

= |v/ (S -*M-ra ) (S-M4m+1 )

x v M-m,M'-m'+l.

Hence one finds

(Mexb = - Z C(LSJ;m,M-m)G(lSJ;m',M'-m' ) ram'

xJ(S«-m)(S-M-hn+l) ^-m,M'-m'+l

x (same integral) . ($7.3)

U. X jj+1 = J If N+l = . A similar calculation gives that

(^ex)li = - 2 C(LSJ;m,M-m)C(LSJ5m',M'-ml ) mm1

xj (S-4Hm)(S-tM-m+l) 5 M-m,M'-m'-l

x (same integral) . (57.U)

The integral involved in Equation (57) will be approximated as

follows. The conduction electrons are the 6s and 5d electrons, of -which

only the s electrons penetrate appreciably into the ion core. So one

may neglect the effect of the 5d electrons and take ug(r) of the Bloch

wave to be spherically symmetrical, i.e., ug(f) = u%(r). The phase

factor exp(ik«iyj can be expanded into a multipole expansion

expdkTi )=4JrZiJ,jjl(tel )T*m(k)Vn ) .

31

If the radius of the Uf shell is small compared with the wave length of

the conduction electron, only the leading term will be of significance.

If one is concerned with the effect of the leading term only, the

integral reduces to

(1,2,--M,—Dag.tifra^tk'rma.) . . J I i N+l I

x ^ Iffi(l#2,.-.N+l,."N)uk(ri)j0(kri)dri..' di^+1 .

The orbital wave functions may be expanded according to Equation (U7),

*/- la» = I (1,2, • • -i-l,i+l, • • -N) (^.)

^ Lm = 2 (^+l) y- r

Then the integral can be written as

( )ug,(rH+1)j0(k,rN4.1) _L ! N+1i'

4f+l"

The last integral is evaluated by standard techniques and the rssult is

Kk.k') = j R*(r1)R( +l)"k.(->|+l)%(ri)

^o(k'4l+l)jo(b-i)^TF[ rir,Hldrldrl!+l • <58>

32

Thus the orbital integral in (57) becomes

Z <$/, 4,)lx%k') = S m.mI(k,k») .

The last result follows from the orthogonality properties of V/ and

'Y' XJH'• Substituting this result into (57)» one obtains

(Mex^l = -2 C2(I£J;m,M-m)(S4M^)l(k,k') yy, ,

(Mex)2 = -2 C2(LSJ;m,M-m)(S-M-ta)l(R,k') ,

m

(^6X^3 ~ ~2 C(LSJjm,M-ni)C(LSJ$m,M-m-l) m

xJ (S4M-m)(S-M-hn+l)l(k,k' ) MjM,+1 ,

(Mex)^ = -2 C(LSJjm,M-m)C(LSJjm,M-m+l m

Xy(S-M-ta)(S«na+l)l(k,k«) S". (59)

By using the wave function (i;5) it is easy to verify that

SZ^JM = 2 C(LSJ;m,M-m)(M-m)Y'imf S,M-m . m

Therefore

<JM' I Sz I JM ) = 2 c2(LSJ.m,M-m)(MHn)^* H, . m

Similarly

33

<( JM' I S_ I JM ) = Zo(l.SJ:m,M^)C(I5J;M,M._NI-l) m

x J(S-tM-m)(S-ÏHm+1 ) f ,

( il' | S+ | JM y = 2. C(l£Jjm,M-m)C(LSJ;m,M-m+l) m

x J (S-M+m)(S*ttî-ffl+l) &u ui_i

Hence the matrix elements in (59) can be written as

(Mex)l = -< JM' | S4SZ j JM >I(k,k') ,

(MeX)2 = - < JM« | S-S2 / JM > I(k,k* ) ,

(MeX)3 = -< JM» / S_| JM>I(k,k'),

(MQX)^ = - < JM' I S+ | JM > I(k,k' ) . (60)

In terms of the spin operator "s of the conduction electron the above

expressions can be put in a compact form

Mgx = -< JM' % ' | S + 2s-s| JM %y I(k,k') . (6L)

Therefore if one is interested only in the spin dependant part, the

Coulomb exchange interaction can be represented by the operator

H = -2I(k,k')s.S*

which is just the de Gennes Hamilton!an (28) for an ion at R = 0. The

strength of interaction A(k,k') is equal to 2l(k,k') and is a function

3U

of the sizes of k, k' only.

In the case of N > 2 Jl +1 one needs to break up the sum in

Equation (SU) into two parts,

Mex = ^ + V&

where represents the interaction with the paired electrons and

is the interaction with the unpaired electrons. Since each paired

electron has equal probabilities to be in the spin up and spin down

state, it can be readily verified that the part M^is independent of

the spin state of the conduction electron. The calculation of is

exactly in parallel with the case of N < 2 JL +1. To avoid repetitions

the final results are given here -without proof

MgX = - <( JM' X • | N/2 42s'S | JM X} I(k,k') . (62)

The spin dependent part of the interaction is given by the same

Hamilton!an (28).

"Within the manifold of constant J value, one may apply the pro-

—

jection theorem and replace S by its projection along J, so

H = -

= - 2I(k,k')(g-l)s.J (63)

where g is the Lande g-factor. If transitions between states of different

J values are considered, the operator in (28) should apply.

35

Discussion

The s-f interaction is believed to be responsible for the following

phenomena in rare earth metals;

1. The ferromagnetic or antiferromagnetic coupling between ions.

2. The anomalous resistivity.

3. The reduction in superconductive transition temperature of

dilute solutions of rare earths in lanthanum.

The first effect is observable in the elements between gadolinium

and thulium. The paramagnetic Curie temperatures of these metals

satisfy the Neel formula (36). That the elements cerium, praseodymium,

neodymium, and samarium are paramagnetic or very weakly anti-

ferromagnetic seems to be an inconsistency in this model. However, it

is known that these elements have peculiar crystal structures instead

of the hexagonal closed packed structure of other elements (Spedding

et al. 195?)» As a result their conduction band structure may be so

different that the same exchange integral I(k,k') no longer applies to

them. The results of Hall measurements on rare earths seem to give seme

support to this speculation (Anderson and Legvold, 1958 a; Kevane et ale

1953). It was found that cerium, praseodymium and neodymium have

positive Hall constants; gadolinium, dysprosium, erbium, and thulium

have negative Hall constants in the paramagnetic temperature region;

and samarium has a very peculiar dependence of its Hall constant on

temperature. The Hall constants for terbium and holmium have not been

measured.

The anomalous resistivity in gadolinium -was explained by Kasuya

36

(1956 a) and by de Germes and Friedel (1958) on the basis of s-f

interaction. This effect is also observable in other rare earths having

more Itf electrons than gadolinium. The dependence of the anomalous

resistivity in the paramagnetic temperature region on the number of Itf

electrons of these metals was discussed by Anderson and Legvold (1958 b)

and by Brout and Buhl (1959). The experimental dependence is in good

accord with the assumption of s-f interaction. Again this effect is

not observed in cerium, phraseodymium, neodymium and samarium. This

also shows that the interaction does not exist or is very weak in these

pure elements.

The negative value of the Hall constant of lanthanum (Kevane et al.

1953) indicates that it has a normal conduction band. TShen small

amounts of rare earths atoms are dissolved in lanthanum, one expects

the s-f interaction to exist between the free electrons and the solute

ions. This effect manifests itself as a reduction of superconductive

transition temperature compared with pure lanthanum. The theory of Suhl

and Matthias (1959) is in satisfactory agreement with the experiment

for almost all rare earth solutions.

One may therefore conclude that a large number of electric and

magnetic properties of rare earths can be understood by a simple s-f

interaction. A detailed study of the conduction band structure of

these elements may give explanation to the exceptional cases as well.

37

INTERPRETATIF OF THE MAGNETIC PROPERTIES

OF DYSPROSIUM

This work was done in collaboration with D. R. Behrendt and S. Legvold

The magnetic properties of poly crystalline dysprosium were measured

by Trombe (19^5, 1951, 1953), and by Elliott et al. (195U); they found

it to be ferromagnetic below 85° K, antif err (magnetic between 85° K and

179° K, and paramagnetic above 179° K. Behrendt et al. (1958) succeeded

in growing a single crystal and measuring the magnetic properties as a

function of direction in the crystal in all three temperature regions.

They found the material to be highly anisotropic, such that the

spontaneous moments lie always in the basal plane. Above 110° K the

dysprosium is isotropic in the basal plane but below that temperature

it has a sixfold ani sot ropy with 1120 ) direction easy. (This is the

direction along the line joining two next nearest neighbors in the

basal plane.) In their paper they gave the experimental results for

the magnetization as a function of temperature, applied field, and

direction in the basal plane for both the ferromagnetic and anti-

ferromagnetic regions.

Neel (1956 a,b) proposed a very interesting, partly phenomenological,

theory to explain the properties in the ferromagnetic and antiferro-

magnetic regions. The content of this theory has been reviewed earlier

in this work. However, Neel's original calculations do not apply for

dysprosium because the single crystals showed properties inconsistent

38

•with his initial assumptions. In fact, between 110° K and 179° K a

unique axis of magnetization does not exist because all directions in

the basal plane are equivalent magnetically.

Above 110° K the isotropy in the basal plane suggests that the inter­

action between the sublattices depends only on the angle between their

magnetic moment vectors. Thus instead of Neel's magnetociystalline

energy terms, one may assume a two term Fourier expansion for the inter­

action energy E% = acos(fA -<^B) + bcos 2(#& -fe). The second term is

necessary to explain the antiferr (magnetic transition. This assignment,

though not unique, does give a consistent fit with the experimental

magnetization curves at all temperatures and orientations.

Basic Equations

The following expression is postulated for the angular dependence

of the magnetic interaction energy per unit volume of the material:

E = - (l/2)25Hcos<t>i - (1/2)MÎCOS^B +A COS(^A-^)

+ b cos (2^ - 2 (1/2)k cos6^-(1/2)k cos 6^B. (6i|)

Here the subscripts A, B refer to the two sublattices; (1/2 )M is the

magnetic moment per unit volume of each sublatticej H is the internal

magnetic field in either the <( 1010 )> or <( 1120 )> direction; and

are the angles between the sublattice magnetizations and the magnetic

field, a, b are the interlattice interaction constants; k is the

anisotropy constant; and the plus sign applies when H is in the 1010^

direction and the minus when H is in the 1120/» direction.

39

For most purposes the dependence of M on H may be ignored. An

estimate of this dependence can be made from the Weiss formula

M = Ms/f(/v/kT)(H+JM)j , (65)

where Ms is the saturation magnetization, pf is the Langevin function,

is the magnetic moment per ion, and }\ is the molecular field constant.

From the paramagnetic susceptibility data, Behrendt et al. (1958) one

estimates to be 10.6 Bohr magnetons and ^ to be U70. The dependence

of M on H is found to be negligible except at temperatures higgler than

130° K. The Weiss formula is used below to give M(H) at these higher

temperatures when the magnetization is parallel to the field and other­

wise the dependence of M on H is ignored.

The parameters M, a, b, k (and Mg at temperatures above 130° K) are

to be chosen to give agreement with the experimental data at each

temperature.

First the temperature range 110° K to 179° K will be considered.

The material is isotropic in the basal plane so one can put k equal to

zero. Equation (6U) can be written as

E = - MH cos(l/2)(<£A+^B)cos(l/2)(<£A-^B)

+ a cos(<£A-<^B) + b cos2(^A- b) , (66)

and then the minimization is easily carried out by regarding 0B

and <f> as the independent variables. Physically it is clear that

only 0 < <^A <: (1/2) 77 and -(l/2)7r ^ 0 need be considered.

From the dependence on A+ B one sees that, at the minimum,

itO

$ k = - » (67)

so the problem reduces to finding the minimum of

E = -MHcos^A+ a cos2<^A+b cosh^. (68)

Depending on the size of H, E as a function of fi^ may have either of

the dependences shown in Figure 2. For small field H, case <X applies

and <p at the minimum of E is near 7T/2 j for large field, case /?

applies and ^ A at minimum energy is zero. There is a discontinuity in

the equilibrium value of f as a function of H at the point where

the two minima are at equal values of E. As long as the minimum is so

near ^ A = 7T/2 that an expansion for small cos 5^A applies, one finds

that at weak fields the equilibrium value of ^ is given try

cos</>A = MH/(Ua-l6b), (69)

and that the discontinuity takes place at the field value given by

M2H2- 8(a-ltb)MH+l6a(a-ltb) = 0 . (70)

The net magnetization of the material CT is given by

(T = (1/2)M cos^A+<l/2)M cos^g , (71)

•where ^k> ^ B are the angles at which E is a minimum. Equations for

the magnetization curves are then

(T = M2H/(Ua-l6b) (72)

for weak fields, Equation (65) for strong fields, and with a

hi

0

0

<ÊA

Figure 2, Typical dependences of E on ^ as given by Equation (68).

h2

discontinuous jump between at a field value given by Equation (70).

In Figures 3 and U the values of M, a, and b -which give the best agree­

ment between theoretical and experimental magnetization curves are

plotted and the actual agreement for three of the magnetization curves

is shown in Figure 6(A). The solid lines are the theoretical curves

and the marked points show the experimental results from Behrendt et al.

(1958).

Next the temperature range 8£° K to 110° K will be considered. The

problem is now more complicated because of the anisotropy. "When the

magnetic field is in the ^ 1010.) direction, Equation (6U) can be

rewritten as

E = -MHCOS(1/2)(<£a+^b) COS(1/2)(^a-^b)

+ a cos(^A-fB) + b cos2(<£a-^b)

+ k cos3(f A+FB) COS3(^a-^b) . (73)

There are several qualitatively different types of minimum energy

configurations possible, depending on the relative sizes of the para­

meters . The one that leads to agreement with the experimental data has

the sublattices oriented antiparallel before the transition and parallel

after. An analysis similar to that of the previous paragraph gives the

following equations for the net magnetization:

CT = M2H/(l;a-l6b+36k) , (7lta)

MH • 36k [ (0~/te)-(l6/3)( cr/M)3+(l6/3)( 0"/m)5 ) f (7Ub)

It3

cr = M , (7Uc)

where the first equation applies for weak fields, the second for inter­

mediate fields, the third for strong fields. An analytical formula for

the field at the transition from Equation (7Ua) to Equation (7i*b) would

be complicated: the actual values were found numerically in each case.

The transition from Equation (7itb) to Equation (7ltc) takes place at

MH = 36k. Figures 3, U, and 5 exhibit the values of the parameters chosen

and Figure 6(B) shows the agreement with the experimental data at 100° K.

The next consideration is the temperature range 85° K to 110° K,

with the field in the (1120 direction. Equation (6b) applies with

the minus signs. One can see that the simple relationship

* k = -*B

does not hold because it would imply that at small field the magnetic

moments are perpendicular to the field in the hard direction of

magnetization <( 10Ï0. In the absence of an external field, the

spontaneous moments lie in < 1120 directions so that the equilibrium

conditions are

f A - =7r' 9/r 0, ± (1/3)77-, ± (2/3)77-, 7r.

The first condition gives the minimum exchange energy and the second the

minimum of anisotropy energy. When k is large enough the system is at

i* A = (1/3,)7Ti B - -(2/3)7T (or a physically equivalent

orientation) for small H, and then there is a discontinuous jump to

hk

1500

Tc2 200

T .TEMPERATURE IN °K

Figure 3» Einpirical values of sublattice magnetization at zero external magnetic field*

to

50 TC| 100 150

X TEMPERATURE, IN eK

TC2 200

Figure U. Empirical values of interaction energy parameters a and b, as introduced in Equations (66) and (73)*

U6

50 TCi 100 150 TC2 200

T, TEMPERATURE IN °K

Figure 5. Enpirical values of the anisotropy constant far the sixfold anisotropy of a sublattice in the basal plane*

47

CO h-z ZD 3 CO (D O

IO O 2

Z

z o 1 h~ < N h-LU Z 0, o (

<

b"

SE

120 °K I40°K-

I70°K -

8 10

H, INTERNAL MAGNETIC FIELD IN KILOGAUSS

( A )

Figure 6. Comparison of typical theoretical and experimental magnetization curves*

48

(Z) h-

CO O O

"o 2

g

ISI

h-LU

b"

I-

1

T IOO°K

H IN <M?0>-

A A A H IN <ldTO>

2 4 6 8 10

H, INTERNAL MAGNETIC FIELD, IN KILOGAUSS

(B)

Figure 6* (Continued)#

it?

P

CO

O 3

10 O

g

S N

UJ z o <

b °0

80°K

•o— H IN <1120>

-A- H IN <I0T0>

10 2 4 6 8

H, INTERNAL MAGNETIC FIELD, IN KILOGAUSS

(C)

Figure 6. (Continued),

5o

<f>k=0,<f>B = 0 when H exceeds a certain value. One can discuss

the magnetization curves by making expansions about these configurations.

In terms of the angles S and é , defined by

* A = T + é '

the energy before the jump is

E =-a+b-k+(l/U)/3MH +(l/2)(a-ltb-t9k)é2

+ (l/8)MïSV + (9/2)kS2 (75)

•where higher powers of S and 6 have been disregarded. Treating S and

é as independent variables, one finds that at minimum energy

<= = -/3MHA(a-Ub49k) ,

S = +yj(MH)2/288k(a-lib-»9k) .

The net magnetization is

IT = (1/2 )M cos A + (1/2 )M cos j>B

= - M(sin(é/2) )j(l/2)JJ cos(£/2) + (l/2) sin(£/2)/.

To the first order in H one finds

0" = 3M2H/l6( a-lib-*9k ) , (76)

and this gives the initial slope of the magnetization curve. Similarly

for high fields, one finds that the first order solutions for the

equilibrium position are

51

A = = 0, 0" = M (77)

The transition takes place at the field

H = 2a/M . (78)

Figure 6(B) shows the comparison of the theory with the experimental

data.

In the ferromagnetic temperature range, T<85° K, the constant a

in the leading term of interlattice interaction energy is negative. The

total energy is a minimum when

where the proper sign of the anisotropy energy should be chosen according

to thé direction of the magnetic field as explained under Equation ( 6 k ) »

Equation (71) becomes

If the field is applied in the easy direction, the material simply remains

magnetized and 0" = M. In fact, a finite field is required to magnetize

the sample; this is probably an effect of domain structure. If the

and Equation (6U) becomes

E = -MHcos + a + b + k cos6 (79)

field is applied in the <( 1010 )> direction, the plus sign in Equation

(79) is to be used. The resulting magnetization curve is

52

MH = 3 6 k [ ( C T / M ) - (16/3)( 07 )3 + (16/3)( cr/M)* j (80a)

<J" = M, (80b)

the saturation taking place at

MH = 36k (81)

Figures 3 and 5 show the choice of M and k which gives best agreement

with the data and a comparison of magnetization curves is given in

Figure 6(C).

In addition to the magnetization curves, this model can be used to

discuss the large anisotropy relative to the c axis. Assuming the field

H is applied in the c direction and considering the antiferromagnetic

region, one may take the energy to be

E = (l/2)MH COS 8k - (1/2 )MH cos 5 g + a cos ( 0 A"*" )

+ b cos 2 ( 9 A+<?B) + (lA)k' cos 2 0 k + (l/2)k»

cos2 6-q , (82)

where 6 9-q are the angles between the magnetization vectors of the

sublattices and the c axis, and k' is the anisotropy constant for this

direction. Minimization of this energy for 0 B near "TT/2 leads to

the susceptibility

J = M2/4(kr + a + Ub) . (83)

The value of k1 can be determined from the previously established values

of M, a, b and the measured susceptibility. The same formula with a, b = 0

53

applies in the ferromagnetic region. The results are displayed in

Figure 7.

The curve of the sublattice magnetization "when plotted as a function

of temperature (Figure 3) resembles that obtained from the conventional

molecular field theory. That the curve goes smoothly through the

ferromagnetic to antiferromagnetic transition temperature Tc gives

support to the basic assumptions that the sublattices are ferromagnetic,

but their interaction may create ferrcmagnetism or antif err omagneti sm.

The antiferromagnetic to paramagnetic transition temperature is the

Curie temperature of a sublattice.

Figure 1| shows the temperature dependence of the interlattice

interaction energy constants a and b. The strong temperature dependence

of these parameters shows that the interaction is different in nature

from the ordinarily assumed exchange energy,

where X is almost temperature independent.

Figures 5 and 7 give the sublattice anisotropy constants as

functions of temperature. The solid curves show the dependence predicted

by Zener's theory (Zener, 1954; see also Keffer, 1955; Pincus, 1959) for

a ferromagnetic lattice. In this theory the anisotropy constant depends

on the temperature through the relationship

Discussion

A' »

k(T) > (84)

5k

> Q_ O » I (Z S h- O

1 o O 1 <2 (/) z E> < O:

LU Û _) o

• o

1 o z $ 1-H z < — M H CO

CVI z CVI o s o

200

T, TEMPERATURE

Figure 7. Qnpirical values of the anisotropy constant for the twofold anisotropy of a sublattice relative to the c axis*

55

where n is an integer. The agreement is good if n = 6 for k and n = 2

for k'. The dependence of the sixfold anisotropy constant implies

a spatial dependence of the anisotropy energy of the form (1/2)k cos ô

sin 6 . This term does not appreciably affect the discussion of the

magnetization curve when the external field is out of the basal plane

because k«k'. The agreement of k' with M3 shows that the assumed

spatial dependence of this anisotropy energy is correct. However, one

can not expect a very good agreement at low temperatures because k1

is so strong that it can no longer be treated as a perturbation to the

spin-wave system (Pincus, 1959)# The application of a, b terms out of

the basal plane also introduces some small corrections on k'. However

the shape of the magnetization curves is too insensitive to the effect

of the a, b terms to tell definitely whether the use of these tenns out

of the basal plane is sensible or not.

The specific heat of the dysprosium was measured by Griff el,

Skochdopole, and Spedding (1956). The magnetic contribution to the

specific heat can be found by subtracting the lattice and the electronic

contributions from the experimental data as described in their paper.

The resulting data are shown in Figure 8, together with the curve (in

dashed lines)

CM = d l/U) J M2]/dT , (85)

as suggested by the molecular field theory and using magnetization

values from Figure 3. As before, A is taken to be 470. The effects of

the interaction between sublattices, as estimated by the values of a and

b, are negligible. The agreement is fair.

56

h- LU

1 I

50 Tci 100 150 TC2 200 T, TEMPERATURE IN °K

Figure 8. Magnetic contribution to the specific heat-»

57

The anomaly at the ferro-antiferromagnetic transition temperature

can be understood by the follovdng thermodynamic argument. The Gibbs

free energy of the system is

G = U - TS + PV - HI , (86)

where I is the net magnetization. Hence

dG = (dO - TdS + PdV - Hdl) - SdT + VdP - IdH

= - SdT + VdP - IdH . (87)

The quantity in the parenthesis is zero by definition. In terms of

quantities per mole

dg = - sdT + vdP - idH . (88)

At the transition the Gibbs function is continuous, so

Sf = Si

So dgf = dg (89)

along the equilibrium surface. The subscripts denote initial and final

values. If the transition is of the first order, then s, v, i, do not

change continuously. But because of Equation (89), one has

-SfdT + VfdP - ifdH = -s dT + v dP - ij_dH .

There is no apparent change in volume during the magnetic transition, one

obtains

58

Su -s. = (i. - if)ÊU fJdT|P • (90)

The initial state is ferromagnetic with ij_ = M; the final state is

antif erromagnetic with no net magnetisation (i = 0), so

4 s = "f I?

Experimentally (Behrendt et al., 1959) it was found that

|p = 1.3xl02 gauss/°K,

M = 5*7xlcA cgs/mole,

M |p - 7.6x10 ergs/°K mole

dT |P

so

The latent heat of transition is then

T A s = 15 cal/mole .

This should correspond to the area under the first peak of the specific

heat curve. The experimental value as estimated from Figure 1 of

Griffel et al. (1956) is about 5 calories per mole.

KLttel (I960) suggested a possible origin of the interlattice

exchange energy postulated in Equation (6i|). A more general theory was

also proposed by Behrendt (1959). Kittel assumes that the strength of

exchange coupling between the sublattices is a linear function of a

lattice parameter (X , and it crosses zero at Of = CYc . The exchange

energy per unit volume is then

59

~ P ( CX - Of c) .

iîhen "writing the free energy of the system, the elastic energy must be

taken into account since a variation in lattice parameter is accompanied

by a change of elastic energy. Hence the free energy per volume is

F = 1/2R( a - a T)2 - p (cx - (X (91)

•where R is the appropriate stiffness constant and 0(T is the equilibrium

value of Of at the temperature T -when MA -1 Mg. The value of CX that

minimizes F is found to be

01 = a T + f . (92) R

If the crystal undergoes a ferro-antiferromagnetic transition, the

change in Of will be

4 Of = M2 e (93)

Experimentally no decisive discontinuous change in either aQ or c0

parameter was observed at the transition (Banister et al. 195U) due to

the large uncertainty of the data.

The minimum value for F is

F = "* |R (FA'MB> " t A T " . (9k)

In terms of the angles, F can be put in the fora

60

+ constant (95)

"which is just the exchange energy expression postulated for dysprosium.

The effective sign of the exchange interaction is determined by

p ( CX CX. A transition of magnetic state "with temperature

occurs when this quantity changes sign. The quantity b in the exchange

energy expression in Equation (6U) corresponds to

However, the quantity on the light-hand side of the above equation

should not have the strong temperature dependence as shown in Figure ii.

Kittel also predicted a pressure dependence of the ferro-

antiferromagnetic transition temperature. By Clausius-Clapeyron equation

b W

A 7 _ V d#/# (96) dP ^ s o G/a T )P

Neglecting the effect of the atmospheric pressure

{JJA F), 3

P

From Equation (9k) one finds

A F = -2 f> M2( CX T- a c)

Hence

61

(^F)p = -2f

Substituting this result and Equation (93) into Equation (96), one obtains

dTcl = 1 1 ~Br o( R Oof T/aT) X . (97)

By the Gruneisen relation, the coefficient of thermal expansion

P- «7 =

where K is the compressibility, 0^ is the lattice specific heat per

volume and if is the Gruneisen constant. K is related to R by

K - 1/ a 2R .

Therefore

s r<i

and

dTci _ 1

•f - — • ( 9 8 )

Assuming a Debye temperature of l£8° K one estimates

Gy "= U.9 cal/ mole °K

at the transition temperature of 85° K. This gives

62

CL = 0.25 cal/ cm3 oK

Taking Y — 2, one obtains

did -5 _ -gp- = 2 cm °K/ cal. = 0.05 K/ atoms .

Experimentally Swensori and Legvold# found this quantity to be -0.0012 +

0.0003 degree Kelvin per atmospheric pressure. This is in definite

disagreement with the theoretical estimation.

Therefore, though the theory of Kittel gives the correct form of

the exchange energy, it does not tell the whole story because seme of

the conclusions do not agree with the experiment.

It is difficult to see at the present how should one proceed to

derive a fundamental theory of the ferro-antif err omagneti c transition

in dysprosium. A detailed study of the magnetic structure in the anti-

ferromagnetic temperature region should be very helpful.

*C. A. Swenson and S. Legvold, Department of Physics, Iowa State University of Science and Technology, Ames, Iowa. Information concerning the pressure dependence of the magnetic transition in dysprosium. Private communication. I960.

63

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66

ACKN OELEDGEjvBÏTS

The author "wishes to express his sincero gratitude to Professor

R. H. Good, Jr. for his constant guidance and encouragement during the

course of this research, to Professor S. Legvold for many stimulating

discussions, and to the Minneapolis-Honeywell Regulator Company and the

International Business Machine Corporation for granting him the fellow­

ships during the academic years of 1958-1959 and 1959-1960.